Geometric Facts about The Mahoujin
Mahoujin In Ep. 48 and Ep. 50, there is a "Magic Circle" shown Akazukin Chacha, as Seravi said, it's called "Mahoujin" in Japanese . Mahoujin and number "7" As we can see, there are "7" small circles on the rim of main circle, aka a heptagon. And as this pic shown, if we draw an externally tangent square on the "inner" circle, then the four vertices of that square just touched the "Main" circle. That's said, the radius ratio of the main circle to the inner circle is: 1:\dfrac {1} {\sqrt {2}} . And, if we make externally tangent circle on the yellow heptagram (not its white rim), than the ratio of this circle to the main circle is 0.624 : 1, aka: \cos \left( \dfrac {360} {7}\right) :1 that means if we make an (wider) heptagram (radius = 1) like this pic: then the radius of its inscribed circle of the inner heptagon is 0.624 that's said, if we draw lines between every vertex of the yellow heptagram n and n+2, than those lines can form a wider heptagram, and this new heptagram's every vertex touches the main circle. We call the externally tangent circle of the yellow heptagram "P circle". Now we calculate the ratio between "inner circle" and "P circle", we find it is: 1:\sqrt {2}\times \cos \left( \dfrac {360} {7}\right) = 1: 0.8817~ let K = 0.8817, then 2 \times \arccos \left( K\right) = 56.29216~ and approximately:(accuracy = 99.933%) 56.29216~ \approx \arctan \left( \dfrac {3} {2}\right) thus, based on the inner circle, we can make both "main circle" and "P circle" using ruler and compass, like this pic: and if we make a tangent line S on the P circle, than it's a nearly perfect heptagon approximation. (the two points where S cross the main circles is 2*(360/7)) Akazukin Chacha does have delicate properties like this. As e can see the thing above, that is the proof of that" 1:\dfrac {1} {\sqrt {2}} " ratio dose relate to the "7" or "heptagon". Now, if we calculate the area of the 7 small circles and the inner circle, we their radius: radius of small circles = \dfrac {1-\sqrt {2}} {2\sqrt {2}} = t thus the area is t^{2} *pi, approximately \dfrac {3} {140}\pi (accuracy = 99.916%) and the area of the inner circle is 0.5(1/2), and the area between every small circles (as the pic's blue area) is (((1-0.5)/7)-(3/140))*pi = 7pi/140.= pi/20. therefore, if we let the area of "Main circle" = 140pi, then the simple integer ratio of inner circle : small circle : left area (blue area) is = 70 : 3 : 7 (accuracy = 99.916%) Again, that is the proof of the area's ratio is related to "7". The "Mahoujin Garden" and "24" In Ep. 48, as we can see the "Garden" where the Mahoujin land, its radius is 5 times of the Mahoujin, as the pic: thus the garden's area = 25*(mahoujin area). and the "meadow" area is (garden-1*Mahoujin) = 24 times of the Mahoujin's area. that's said, every fan-shaped area (as the pic, divided by red straight lines) is equal to a Mahoujin. Is "24" meaningful in Akazukin Chacha? As we can see, "Princess Medallion" has 24 "golden beans" around the gem. And the LCM of Medallion's gem (hexagon, 6) and Chacha's ring's gem (octagon, 6) is also 24. Are those just coincidences? maybe not so. Category:Extras